The F-method and a branching problem for generalized Verma modules associated to $({\mathrm{Lie~}G_2},{\operatorname{so}(7)})$

Todor Milev and Petr Somberg

Mathematical Institute of Charles University, Sokolovská 83, Praha 8 - Karlín, Czech Republic
Department of Mathematics, University of Massachusetts Boston, 100 William T. Morrissey Boulevard, Boston, MA 02125, USA


Abstract: The branching problem for a couple of non-compatible Lie algebras and their parabolic subalgebras applied to generalized Verma modules was recently discussed in [15]. In the present article, we employ the recently developed F-method, [10], [11] to the couple of non-compatible Lie algebras $\mathrm{Lie~}G_2\stackrel{i}{\hookrightarrow }{\operatorname{so}(7)}$, and generalized conformal ${\operatorname{so}(7)}$-Verma modules of scalar type. As a result, we classify the $i({\mathrm{Lie~}G_2}) \cap {\mathfrak{p}}$-singular vectors for this class of $\operatorname{so}(7)$-modules.

AMSclassification: primary 22E47; secondary 17B10, 13C10.

Keywords: generalized Verma modules, conformal geometry in dimension 5, exceptional Lie algebra {\mathrm{Lie~}G_2}, F-method, branching problem.

DOI: 10.5817/AM2013-5-317